Inversion of up and down going signal for ocean bottom data

Complete modeling

To model beyond primaries and first order receiver pegleg multiples, one can use the separation operator that maps over-under data to up-going and down-going data,

$$\left[ \begin{array}{c} \bold L_{\uparrow} \\ \bold L_{\downarrow} \end{array} \right] = \bold S_{ou} \bold A ,$$

where $\bold A$ represents the wave equation forward modelling operator that generates over and under signals. $\bold S_{ou}$ is a separation operator that extracts the up- and down-going signals from over-under data. In matrix form, the inversion scheme has the following fitting goal:

$$0 \approx \left[ \begin{array}{c} \bold L_{\uparrow} \\ \bold L_... ...pz} \left[ \begin{array}{c} \bold d_{p} \\ \bold d_{z} \end{array} \right] ,$$

where $\bold S_{pz}$ is a separation operator that extracts the up- and down-going signals from PZ data. Note that $\bold d_{\uparrow}$ and $\bold d_{\downarrow}$ can be viewed as processed data from the orignal recorded $\bold d_{p}$ and $\bold d_{Z}$. The advantage of this joint modeling is that we are now imaging all multiples event that return to the ocean bottom receivers going upward or downward.

For our complete modeling operator, $\bold L_{\uparrow}$ and $\bold L_{\downarrow}$, there is an alternate way to interpret the inverson problem we have set-up from above. Consider an equivalent fitting goal below,

$$0 \approx \bold S^{-1}_{pz} \bold S_{ou} \bold A \bold m - \left[ \begin{array}{c} \bold d_{p} \\ \bold d_{z} \end{array} \right].$$

The above fitting goal converts our model into over-under data. Afterward, over-under data are separated into up-going and down-going data. Finally, the up-going and down-going data are converted into PZ data using $\bold S^{-1}_{pz}$. Therefore, this inversion scheme can be interpreted as fitting both P and Z data using only acoustic equation.

Inversion of up and down going signal for ocean bottom data